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Full text of "Mathematical And Physical Papers - Iii"

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ON THE  MOTION  OF  PENDULUMS.
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moment of the resisting force was proportional to the fourth power of the radius. From these laws Coulomb concluded that each small element of any one of the disks experienced a resistance varying as the area of the element multiplied by its linear velocity. It should be observed that Coulomb was only authorized by his experiments to assert this law to be true in the case of oscillations of given period, inasmuch as the time of oscillation was nearly the same in all the experiments.
Let a be the radius of the disk in the fluid, r the time of oscillation, 9 the angular displacement of the disk, measured from its mean position, I the moment of inertia of the whole system ; and let 1 : 1  m be the ratio in which the arc of oscillation is diminished in one oscillation. According to the formula (15) we have
for the factor which expresses the ratio of the arc of oscillation at the end of the time t to the initial arc. At the end of one oscillation t~T, and the value of the above factor is 1  m, which is given by observation. Putting for /3 its value, in which Mrf = I, and nr = TT, we get
Let T be the time of oscillation, and 70 the moment of inertia, when the under disk is removed: then /=/0r2T~2. Also if M be the mass and R the radius of the large graduated disk, we have /0 = ^MK\ neglecting, as Coulomb did, the rotatory inertia of the copper cylinder. Substituting in (162), we get
loge (1 - m)'1 = SHrTrV/^T^rVJET8 Jf- l ...... (163).
Let W be the weight of the disk in grammes. Then the mass of the disk is equal to that of W cubic centimetres or 1000 W cubic millimetres of water. Hence M = lOOOpTF, a millimetre being the unit of length. Substituting in (163), and solving with respect to V/A', we get V// = 1000 x & log. 10 . TT-* F2T-VM Iog10 (1 - m)'1 . . .(164),
and the same value of *Jp ought to result from different experiments.
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